Let f_i be an infinite sequence of elliptic Hecke eigenforms such that the individual weights and levels are unbounded as i goes to infinity. When does one expect that f_i's have a common zero? Any guess based on heuristics/ conjectures is welcome! Any non-trivial example of this phenoemena?
4
-
$\begingroup$ By "nontrivial" do you mean the list should not admit a finite partition into subsets, each having a common multiple with a zero in the same place? $\endgroup$– S. Carnahan ♦Commented Mar 17, 2013 at 7:16
-
$\begingroup$ If you'll allow me to use oldforms (which you probably won't) then there's a trick to ensure a common zero -- because of elementary considerations involving elliptic points (which can be dressed up via a "stacky" argument to look much more fancy), if the weight of a level 1 form (eigenform or not) is not 0 mod 4 then it will have a zero at i, and if it's not 0 mod 6 it will have a zero at $\rho$. However this trick goes away if you force the conductors to go to infinity, and in this case why would one expect common zeros at all? $\endgroup$– user30035Commented Mar 17, 2013 at 9:14
-
$\begingroup$ Yes, by non-trivial, I mean what Carnahan said and also to avoid oldforms. It seems that this might be a very rare phenomena. However, I do not know of any heuristics or conjectures even hinting this. $\endgroup$– BasicCommented Mar 17, 2013 at 14:08
-
$\begingroup$ Does Borcherd product type consideration say something? $\endgroup$– BasicCommented Mar 19, 2013 at 6:12
Add a comment
|